Limits — Question 8

Question 8

Prove directly from the $\varepsilon$–$\delta$ definition that $$\lim_{(x,y)\to(1,-2)}(3x-4y)=11.$$ Tasks

1. Rewrite the output error.

2. Bound it by the Euclidean input distance.

3. Give an explicit valid choice of $\delta$.

Question 8 – Solution

Strategy. Use Cauchy–Schwarz to compare the linear output change with the distance to $(1,-2)$.

Step 1: Error Let $h=x-1$ and $k=y+2$. Then $$|(3x-4y)-11|=|3h-4k|.$$

Step 2: Cauchy–Schwarz $$|3h-4k|\le\sqrt{3^2+(-4)^2}\sqrt{h^2+k^2}=5\sqrt{(x-1)^2+(y+2)^2}.$$

Step 3: Choose $\delta$ Given $\varepsilon>0$, set $$\boxed{\delta=\varepsilon/5}.$$ If $0<\sqrt{(x-1)^2+(y+2)^2}<\delta$, then $$|(3x-4y)-11|<5\delta=\varepsilon.$$ This is exactly the two-variable limit definition, so the claimed limit is proved.